Branched Latent Neural Operators

We introduce Branched Latent Neural Operators (BLNOs) to learn input-output maps encoding complex physical processes. A BLNO is defined by a simple and compact feedforward partially-connected neural network that structurally disentangles inputs with different intrinsic roles, such as the time variable from model parameters of a differential equation, while transferring them into a generic field of interest. BLNOs leverage interpretable latent outputs to enhance the learned dynamics and break the curse of dimensionality by showing excellent generalization properties with small training datasets and short training times on a single processor. Indeed, their generalization error remains comparable regardless of the adopted discretization during the testing phase. Moreover, the partial connections, in place of a fully-connected structure, significantly reduce the number of tunable parameters. We show the capabilities of BLNOs in a challenging test case involving biophysically detailed electrophysiology simulations in a biventricular cardiac model of a pediatric patient with hypoplastic left heart syndrome. The model includes a purkinje network for fast conduction and a heart-torso geometry. Specifically, we trained BLNOs on 150 in silico generated 12-lead electrocardiograms (ECGs) while spanning 7 model parameters, covering cell-scale, organ-level and electrical dyssynchrony. Although the 12-lead ECGs manifest very fast dynamics with sharp gradients, after automatic hyperparameter tuning the optimal BLNO, trained in less than 3 hours on a single CPU, retains just 7 hidden layers and 19 neurons per layer. The mean square error is on the order of $10^{-4}$ on an independent test dataset comprised of 50 additional electrophysiology simulations. This paper provides a novel computational tool to build reliable and efficient reduced-order models for digital twinning in engineering applications.